A318108
a(n) = Sum_{k=0..n} (3*n-2*k)!/((n-k)!^3*k!)*(-3)^k, n >= 0.
Original entry on oeis.org
1, 3, 27, 303, 3771, 49653, 677979, 9496791, 135572859, 1963940073, 28783474677, 425872190241, 6350923156059, 95341185353781, 1439433069482547, 21839152342265703, 332769145298428539, 5089688869615075521, 78108038975852093889, 1202268428203687094493, 18555675891246972931221
Offset: 0
A(x) = 1 + 3*x + 27*x^2 + 303*x^3 + 3771*x^4 + 49653*x^5 + 677979*x^6 + ...
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Table[Sum[(3n-2k)!/(((n-k)!)^3 k!) (-3)^k,{k,0,n}],{n,0,20}] (* Harvey P. Dale, Mar 01 2019 *)
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a(n) = sum(k=0, n, (3*n-2*k)!/((n-k)!^3*k!)*(-3)^k);
vector(21, n, a(n-1))
A318109
a(n) = Sum_{k=0..n} (3*n-2*k)!/((n-k)!^3*k!)*(-2)^k.
Original entry on oeis.org
1, 4, 46, 652, 10186, 168304, 2884456, 50723824, 909192538, 16538659384, 304391739796, 5655971294824, 105929883322576, 1997228410630912, 37871584674309376, 721672204654077952, 13811327854028171098, 265324110145941691912, 5114208160758838538044, 98874597697991698311832, 1916741738060370782929036
Offset: 0
A(x) = 1 + 4*x + 46*x^2 + 652*x^3 + 10186*x^4 + 168304*x^5 + 2884456*x^6 + ...
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a(n) = sum(k=0, n, (3*n-2*k)!/((n-k)!^3*k!)*(-2)^k);
vector(21, n, a(n-1))
A318105
Triangle read by rows: T(n,k) = (4*n - 3*k)!/((n-k)!^4*k!).
Original entry on oeis.org
1, 24, 1, 2520, 120, 1, 369600, 22680, 360, 1, 63063000, 4804800, 113400, 840, 1, 11732745024, 1072071000, 33633600, 415800, 1680, 1, 2308743493056, 246387645504, 9648639000, 168168000, 1247400, 3024, 1, 472518347558400, 57718587326400, 2710264100544, 61108047000, 672672000, 3243240, 5040, 1
Offset: 0
A(x;t) = 1 + (24 + t)*x + (2520 + 120*t + t^2)*x^2 + (369600 + 22680*t + 360*t^2 + t^3)*x^3 + ...
Triangle starts:
n\k [0] [1] [2] [3] [4] [5] [6]
[0] 1;
[1] 24, 1;
[2] 2520, 120, 1;
[3] 369600, 22680, 360, 1;
[4] 63063000, 4804800, 113400, 840, 1;
[5] 11732745024, 1072071000, 33633600, 415800, 1680, 1;
[6] 2308743493056, 246387645504, 9648639000, 168168000, 1247400, 3024, 1;
[7] ...
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t[n_,k_] := (4*n - 3*k)!/((n-k)!^4*k!); Table[t[n, k], {n, 0, 10}, {k , 0,n}] // Flatten (* Amiram Eldar, Nov 07 2018 *)
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T(n, k) = (4*n-3*k)!/(k!*(n-k)!^4);
concat(vector(8, n, vector(n, k, T(n-1, k-1))))
/*
test:
P(n, v='t) = subst(Polrev(vector(n+1, k, T(n, k-1)), 't), 't, v);
diag(expr, N=22, var=variables(expr)) = {
my(a = vector(N));
for (k = 1, #var, expr = taylor(expr, var[#var - k + 1], N));
for (n = 1, N, a[n] = expr;
for (k = 1, #var, a[n] = polcoef(a[n], n-1)));
return(a);
};
apply_diffop(p, s) = {
s=intformal(s);
sum(n=0, poldegree(p, 'Dx), s=s'; polcoef(p, n, 'Dx) * s);
};
\\ diagonal property:
x='x; y='y; z='z; w='w; t='t;
diag(1/(1 - (x+y+z+w + t*x*y*z*w)), 9, [x, y, z, w]) == vector(9, n, P(n-1))
\\ annihilating diffop:
y = Ser(vector(101, n, P(n-1)), 'x);
p = x^2*(3*t*x + 1)^2*((t*x - 1)^4 - 256*x)*Dx^3 + 3*x*(3*t*x + 1)*((t*x - 1)^3*(6*t^2*x^2 + 3*t*x - 1) - 384*x*(t*x + 1))*Dx^2 + (t*x - 1)*((t*x - 1)*(63*t^4*x^4 + 66*t^3*x^3 - 18*t*x + 1) + 48*x*(15*t*x + 17))*Dx + (t*x - 1)*(t*(9*t^4*x^4 + 12*t^3*x^3 + 6*t^2*x^2 - 12*t*x + 1) - 24*(15*t*x - 1));
0 == apply_diffop(p, y)
*/
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